A Constructive Solution to the Yang–Mills Mass Gap

This work develops a constructive and physically grounded path to the Yang–Mills mass gap using the framework of Unified Substrate Theory (UST). Instead of quantizing classical Yang–Mills fields directly—an approach that encounters problems with gauge fixing, BRST ghosts, and violations of reflection positivity—the method begins with a deeper substrate whose dynamics exhibit intrinsic discrete-scale (β-mode) behaviour. These regeneration modes appear in many independent physical systems and correspond to a well-defined, self-adjoint substrate operator with a strict and robust spectral gap. Gauge fields emerge kinematically from internal SU(3) orientation fields of the substrate, ensuring that gauge symmetry is preserved without introducing gauge fixing or ghost fields. This structure guarantees that all Osterwalder–Schrader positivity requirements are satisfied at the substrate level. The Yang–Mills Hamiltonian is shown to inherit the substrate gap directly, establishing the existence of a strictly positive mass gap. Extensive numerical analysis across a wide range of parameters (regeneration frequencies, potential strengths, dimensional extensions, and lattice resolutions) demonstrates that the substrate spectral gap is universal, stable, and persists in the continuum limit. These results provide a coherent and constructive mechanism for obtaining a Yang–Mills mass gap from deeper substrate dynamics. While the full constructive measure and reconstruction steps required for a formal Clay-level proof are outlined and reserved for future work, this manuscript establishes the spectral, structural, and positivity foundations necessary for such a programme.